Conormal Geometry of Maximal Minors

Authors: Kleiman S.¹; Thorup A.²

Source: Journal of Algebra, Volume 230, Number 1, August 2000 , pp. 204-221(18)

Publisher: Academic Press

Abstract:

Let A be a Noetherian local domain, N be a finitely generated torsion-free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C Proj(A[M]) and r dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]_p is not a finitely generated module over A[M]_p, and denote the preimage of W in C by E. We prove this: (1) dim E = r - 1 if either (a) N is free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex-analytic variety, and applied it to improve a characterization of Thom's A_f-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of Böger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by Thorup, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M]. Copyright 2000 Academic Press.

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, Room 2-278 MIT, 77 Massachusetts Ave., Cambridge, Massachusetts, 02139-4307 2: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, København Ø, DK-2100, Danmark

Publication date: 2000-08-01

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• In this Subject: Mathematics and Statistics
• By this author: Kleiman S. ;  Thorup A.

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